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CHAPTER III

METRIC SPACES

This chapter, together with Chapter V, constitutes the core of this first
volume: in. them is developed the geometric language in which are now
expressed the results of analysis, and which has made it possible to give to
these results their full generality, as well as to supply for them the simplest
and most perspicuous proofs. Most of the notions introduced in this chapter
have very intuitive meanings, when specialized to "ordinary" three-dimen-
sional space; after some experience with their use, both in problems and in
the subsequent chapters, the student should be able to reach the conviction
that, with proper safeguards, this intuition is on the whole an extremely
reliable guide, and that it would be a pity to limit it to its classical range of
application.

There are almost no genuine theorems in this chapter; most results follow
in a straightforward manner from the definitions, and those which require a
little more elaboration never lie very deep. Sections 3.1 to 3.13 are essentially
concerned with laying down the terminology;,it may seem to the unprepared
reader that there is a very great deal of it, especially in Sections 3.5 to 3.8,
which really are only various ways of saying the same things over again;
the reason for this apparent redundancy of the language is to be sought in
the applications: to dispense with it (as one theoretically might) would often
result in very awkward and cumbersome expressions, and it has proved
worthwhile in practice to burden the memory with a few extra terms in order
to achieve greater clarity.

The most important notions developed in this chapter are those of
completeness (Section 3.14), compactness (Sections 3.16 to 3.18) and connected-
ness (Section 3.19), which will be repeatedly used later on, and of which the
student should try to get as thorough a grasp as possible before he moves on.

Metric spaces only constitute one special kind of "topological spaces,'"
and this chapter may therefore be visualized as introductory to the study of



	CHAPTER III METRIC SPACES

	This chapter, together with Chapter V, constitutes the core of this first volume: in. them is developed the geometric language in which are now expressed the results of analysis, and which has made it possible to give to these results their full generality, as well as to supply for them the simplest and most perspicuous proofs. Most of the notions introduced in this chapter have very intuitive meanings, when specialized to "ordinary" three-dimen- sional space; after some experience with their use, both in problems and in the subsequent chapters, the student should be able to reach the conviction that, with proper safeguards, this intuition is on the whole an extremely reliable guide, and that it would be a pity to limit it to its classical range of application. There are almost no genuine theorems in this chapter; most results follow in a straightforward manner from the definitions, and those which require a little more elaboration never lie very deep. Sections 3.1 to 3.13 are essentially concerned with laying down the terminology;,it may seem to the unprepared reader that there is a very great deal of it, especially in Sections 3.5 to 3.8, which really are only various ways of saying the same things over again; the reason for this apparent redundancy of the language is to be sought in the applications: to dispense with it (as one theoretically might) would often result in very awkward and cumbersome expressions, and it has proved worthwhile in practice to burden the memory with a few extra terms in order to achieve greater clarity. The most important notions developed in this chapter are those of completeness (Section 3.14), compactness (Sections 3.16 to 3.18) and connected- ness (Section 3.19), which will be repeatedly used later on, and of which the student should try to get as thorough a grasp as possible before he moves on. Metric spaces only constitute one special kind of "topological spaces,'" and this chapter may therefore be visualized as introductory to the study of